Question: $D(t)$ models the distance (in thousands of $\text{km}$ ) from the earth to the Moon $t$ days after the moon's perigee (when it's closest to the earth). Here, $t$ is entered in radians. $D(t) = -21\cos\left(\dfrac{2\pi}{29.5}t\right) + 384$ How many days after its perigee does the moon first reach $380$ thousands of $\text{ km}$ from the Earth? Round your final answer to the nearest whole day.
Converting the problem into mathematical terms $D(t) = -21\cos\left({\dfrac{2\pi}{29.5}}t\right) + 384$ has a period of $\dfrac{2\pi}{{\scriptsize\dfrac{2\pi}{29.5}}}=29.5$ days. We want to find the first solution to the equation $D(t)=380$ within the period $0<t<29.5$. The answer The equation's two solutions within the desired period (rounded to the nearest whole day) are $6$ and $23$. Therefore, the Moon first reaches $380$ thousands of $\text{ km}$ from earth about $6$ days after its perigee.